# Optimal Timing to Trade along a Randomized Brownian Bridge

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## Abstract

**:**

## 1. Introduction

## 2. Prior Belief and Price Dynamics

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 3. Optimal Liquidation Problems

**Proposition**

**1.**

- 1.
- $G(u,x)>0$, $\forall (u,x)\in [t,\overline{T}]\times {\mathbb{R}}_{+}\Rightarrow {\tau}^{*}=\overline{T}$,
- 2.
- $G(u,x)\le 0$, $\forall (u,x)\in [t,\overline{T}]\times {\mathbb{R}}_{+}\Rightarrow {\tau}^{*}=t$.

**Proof.**

#### 3.1. Stocks

**Proposition**

**2.**

- (i) downward-sloping and concave in x if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le q\left(t\right)-2,$$$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge q\left(t\right)-1,$$
- (ii) downward-sloping and convex in x if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}q\left(t\right)-2\le x\le q\left(t\right)-1,$$
- (iii) upward-sloping and concave in x if$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}q\left(t\right)-2\le x\le q\left(t\right)-1,$$
- (iv) upward-sloping and convex in x if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge q\left(t\right)-1,$$$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le q\left(t\right)-2,$$$$q\left(t\right):={X}_{0}-\frac{\mu {\sigma}^{2}T+(\frac{1}{2}{\sigma}^{2}-r)(t{\sigma}_{D}^{2}+T{\sigma}^{2}(T-t))}{{\sigma}_{D}^{2}-T{\sigma}^{2}}.$$

**Proposition**

**3.**

- (i)
- downward-sloping and concave in t if$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}},$$
- (ii)
- downward-sloping and convex in t if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}},$$
- (iii)
- upward-sloping and concave in t if$${\sigma}_{D}>\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}},$$
- (iv)
- upward-sloping and convex in t if$${\sigma}_{D}<\sqrt{T}\sigma \phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}x\le {X}_{0}-\frac{\mu {\sigma}^{2}T}{{\sigma}_{D}^{2}-T{\sigma}^{2}}.$$

#### 3.2. Call and Put Options

**Lemma**

**1.**

**Proposition**

**4.**

**Proof.**

## 4. Numerical Implementation

## 5. Conclusions

## 6. Proofs

#### 6.1. Normal Distribution

#### 6.2. Double Exponential Distribution

#### 6.3. Proof of Propositions 2 and 3

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Baurdoux, Erik J., Nan Chen, Budhi A. Surya, and Kazutoshi Yamazaki. 2015. Optimal double stopping of a Brownian bridge. Advances in Applied Probability 47: 1212–34. [Google Scholar] [CrossRef]
- Bensoussan, Alain, and Jacques-Louis Lions. 1982. Applications of Variational Inequalities in Stochastic Control. Amsterdam: North-Holland Publishing Co. [Google Scholar]
- Brennan, Michael J., and Eduardo S. Schwartz. 1990. Arbitrage in stock index futures. Journal of Business 63: S7–S31. [Google Scholar] [CrossRef]
- Brody, Dorje C., Mark HA Davis, Robyn L. Friedman, and Lane P. Hughston. 2009. Informed traders. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. London: The Royal Society, vol. 465, pp. 1103–1122. [Google Scholar] [CrossRef][Green Version]
- Brody, Dorje C., Lane P. Hughston, and Andrea Macrina. 2008b. Information-based asset pricing. International Journal of Theoretical and Applied Finance 11: 107–42. [Google Scholar] [CrossRef]
- Cartea, Álvaro, Sebastian Jaimungal, and Damir Kinzebulatov. 2016. Algorithmic trading with learning. International Journal of Theoretical and Applied Finance 19: 1650028. [Google Scholar] [CrossRef]
- Dai, Min, Yifei Zhong, and Yue Kuen Kwok. 2011. Optimal arbitrage strategies on stock index futures under position limits. Journal of Futures Markets 31: 394–406. [Google Scholar] [CrossRef]
- Ekström, Erik, and Juozas Vaicenavicius. 2017. Optimal stopping of a Brownian bridge with an unknown pinning point. arXiv preprint. arXiv:1705.00369. [Google Scholar]
- Ekström, Erik, and Henrik Wanntorp. 2009. Optimal stopping of a Brownian bridge. Journal of Applied Probability 46: 170–80. [Google Scholar] [CrossRef]
- Filipović, Damir, Lane P. Hughston, and Andrea Macrina. 2012. Conditional density models for asset pricing. International Journal of Theoretical and Applied Finance 15: 1250002. [Google Scholar] [CrossRef]
- Guo, Kevin, and Tim Leung. 2017. Understanding the non-convergence of agricultural futures via stochastic storage costs and timing options. Journal of Commodity Markets 6: 32–49. [Google Scholar] [CrossRef][Green Version]
- Hughston, Lane P., and Andrea Macrina. 2012. Pricing fixed-income securities in an information-based framework. Applied Mathematical Finance 19: 361–79. [Google Scholar] [CrossRef]
- Johannes, Michael, and Andrew Dubinsky. 2006. Earnings Announcements and Equity Options. Unpublished Paper. New York: Columbia University. [Google Scholar]
- Karatzas, Ioannis, and Steven E. Shreve. 1998. Methods of Mathematical Finance. New York: Springer. [Google Scholar]
- Kou, Steven G. 2002. A jump-diffusion model for option pricing. Management Science 48: 1086–101. [Google Scholar] [CrossRef]
- Leung, Tim, Jiao Li, Xin Li, and Zheng Wang. 2016. Speculative futures trading under mean reversion. Asia-Pacific Financial Markets 23: 281–304. [Google Scholar] [CrossRef]
- Leung, Tim, and Peng Liu. 2012. Risk premia and optimal liquidation of credit derivatives. International Journal of Theoretical & Applied Finance 15: 1250059. [Google Scholar]
- Leung, Tim, and Mike Ludkovski. 2011. Optimal timing to purchase options. SIAM Journal on Financial Mathematics 2: 768–93. [Google Scholar] [CrossRef]
- Leung, Tim, and Mike Ludkovski. 2012. Accounting for risk aversion in derivatives purchase timing. Mathematics & Financial Economics 6: 363–86. [Google Scholar]
- Leung, Tim, and Marco Santoli. 2014. Accounting for earnings announcements in the pricing of equity options. Journal of Financial Engineering 1: 1450031. [Google Scholar] [CrossRef][Green Version]
- Leung, Tim, and Marco Santoli. 2016. Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation. SpringerBriefs in Quantitative Finance. New York: Springer. [Google Scholar]
- Leung, Tim, and Yoshihiro Shirai. 2015. Optimal derivative liquidation timing under path-dependent risk penalties. Journal of Financial Engineering 2: 1550004. [Google Scholar] [CrossRef][Green Version]
- Macrina, Andrea. 2014. Heat kernel models for asset pricing. International Journal of Theoretical and Applied Finance 17: 1450048. [Google Scholar] [CrossRef]
- Wilmott, Paul, Sam Howison, and Jeff Dewynne. 1995. The Mathematics of Financial Derivatives: A Student Introduction, 1st ed.Cambridge: Cambridge University Press. [Google Scholar]

**Figure 1.**Path simulation of (4) with the prior belief on the future log-price following (

**a**) two-point discrete distribution and (

**b**) normal distribution. Parameters: (

**a**) ${\delta}_{u}=0.3,\phantom{\rule{4pt}{0ex}}{\delta}_{d}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}-0.3,\phantom{\rule{4pt}{0ex}}{p}_{u}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5,\phantom{\rule{4pt}{0ex}}{p}_{d}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5$; (

**b**) $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.3$. Common parameters: ${X}_{0}=1,\phantom{\rule{4pt}{0ex}}T=1,\phantom{\rule{4pt}{0ex}}\sigma =0.4$.

**Figure 2.**$A(t,x)$ under three different distribution with common Mean = 0 and Var = $0.36$ at $t=0.1$ (

**a**) and $t=0.8$ (

**b**). Parameters: (two-point) ${\delta}_{u}=-{\delta}_{d}=0.6,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$; (normal) $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.6$; (double exponential) $\theta =0,\phantom{\rule{4pt}{0ex}}{p}_{1}={p}_{2}=0.5,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=2.357$. Common parameters: ${S}_{0}=2.72({X}_{0}=1),\phantom{\rule{4pt}{0ex}}r=0.1,\phantom{\rule{4pt}{0ex}}\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1,\phantom{\rule{4pt}{0ex}}\sigma =0.4.$

**Figure 3.**Optimal boundaries for selling stock under two-point discrete distribution (

**a**,

**b**), normal distribution (

**c**) and double exponential distribution (

**d**). Parameters for each plot are as follows: (

**a**) ${\delta}_{u}=-{\delta}_{d}=0.1,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$; (

**b**) ${\delta}_{u}=-{\delta}_{d}=2,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$; (

**c**) $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.2$; (

**d**) $\theta =0$, ${p}_{1}={p}_{2}=0.5,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=10$. Common parameters: ${S}_{0}=2.72({X}_{0}=1),\phantom{\rule{4pt}{0ex}}r=0.1,\phantom{\rule{4pt}{0ex}}\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1$, $\sigma =0.4.$

**Figure 4.**Optimal trading boundaries for stock liquidation under the two-point discrete distribution (

**a**,

**b**), normal distribution (

**c**,

**d**) and double exponential distribution (

**e**,

**f**). Parameters: (

**a**) ${\sigma}_{D}=0.2$; (

**b**) $\mu =0$; (

**c**) ${\delta}_{u}=-{\delta}_{d}=0.1$; (

**d**) ${p}_{u}={p}_{d}=0.5$; (

**e**) ${\lambda}_{1}={\lambda}_{2}=10$; (

**f**) ${p}_{1}={p}_{2}=0.5$. Other common parameters are the same as Figure 3.

**Figure 5.**Optimal trading regions for stock liquidation under the two-point discrete distribution. The continuation regions are in yellow (light) color and the exercise regions are in blue (dark), and they are disconnected. Parameters: ${\delta}_{u}=-{\delta}_{d}=0.8,\phantom{\rule{4pt}{0ex}}{p}_{u}={p}_{d}=0.5$. Common parameters: ${X}_{0}=1,\phantom{\rule{4pt}{0ex}}r=0.1$, $\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1,\phantom{\rule{4pt}{0ex}}\sigma =0.4$.

**Figure 6.**Optimal trading regions for stock liquidation under the double exponential distribution. The continuation regions are in yellow (light) color and the exercise regions are in blue (dark), and they are disconnected. Parameters for (

**a**): $\theta =0,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=4.714,\phantom{\rule{4pt}{0ex}}{p}_{1}={p}_{2}=0.5$; parameters for (

**b**): $\theta =0,\phantom{\rule{4pt}{0ex}}{\lambda}_{1}={\lambda}_{2}=3.536,\phantom{\rule{4pt}{0ex}}{p}_{1}={p}_{2}=0.5$. Common parameters: ${X}_{0}=1,\phantom{\rule{4pt}{0ex}}r=0.1$, $\overline{T}=1,\phantom{\rule{4pt}{0ex}}T=1.1,\phantom{\rule{4pt}{0ex}}\sigma =0.4.$.

**Figure 7.**Optimal boundaries for selling European call with ${S}_{0}=105$ (

**a**) and put with ${S}_{0}=95$ (

**b**) under normal distribution. The strike price $K=100$ and maturity $T=1$, and the other parameters: $\mu =0,\phantom{\rule{4pt}{0ex}}{\sigma}_{D}=0.1,\phantom{\rule{4pt}{0ex}}r=0.1,\phantom{\rule{4pt}{0ex}}\overline{T}=1,\phantom{\rule{4pt}{0ex}}\sigma =0.4$. The green line is defined by $G=0$, and continuation region contains $G\ge 0$.

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**MDPI and ACS Style**

Leung, T.; Li, J.; Li, X. Optimal Timing to Trade along a Randomized Brownian Bridge. *Int. J. Financial Stud.* **2018**, *6*, 75.
https://doi.org/10.3390/ijfs6030075

**AMA Style**

Leung T, Li J, Li X. Optimal Timing to Trade along a Randomized Brownian Bridge. *International Journal of Financial Studies*. 2018; 6(3):75.
https://doi.org/10.3390/ijfs6030075

**Chicago/Turabian Style**

Leung, Tim, Jiao Li, and Xin Li. 2018. "Optimal Timing to Trade along a Randomized Brownian Bridge" *International Journal of Financial Studies* 6, no. 3: 75.
https://doi.org/10.3390/ijfs6030075